let $X(t) = Ae^{j\omega_c t}+W(t)$, where $W(t)$ is a gaussian process that follows the statistics $W \sim \mathcal{CN}(0,\sigma^2)$ and $\omega_c$ denotes the carrier pulse frequency and $A$ is a positive constant value.
For $A>\sigma$, $ |X(t)|$ seems to follow a gaussian process, but as said in the comments, for lower $A$ values $|X(t)|$ does follow a Rician distribution. Can someone proof this? Because $Ae^{j\omega_c t}$ could be understood as $Ae^{j\phi}$ where $\phi \sim unif[-\pi,\pi]$, and so it is not a constant value as it is said in the standard Rician distribution definition.
This is the histogram that I obtained with MATLAB for $A=1$ :
My MATLAB code is following:
fm = 10e3; % sampling rate
Tm = 1/(fm); % sampling time
t = (1:1:2e6)*Tm; % time vector
fc = 1e-2*fm; % carrier frequency
A = 1; % Amplitude
x = A*exp(1i*2*pi*fc*t); % signal
sigma = 2.5e-5; % noise variance
w = sigma/sqrt(2)*(randn(length(t),1) + 1i*randn(length(t),1))'; % noise vector
y_th = x+w; % received signal
hist(abs(y_th),200)